Rate (L 34) is first introduced on page 113. We know that rates are fractions. Unfortunately, it is not explicitly stated that you should set a rate up as REDUCED fraction, but you can see in the examples for this lesson that each rate is converted to an equivalent fraction with a 1 in the numerator or a 1 in the denominator. We always write both rates because we could need either one, depending upon the word problem, but we will look at this again in Practice 63a.
You do get a bit of help in Rate Word Problems (L 36) with the “Note” in the middle of the page saying that you can reduce rates, if you want to. For students, it is better to reduce them. I constantly emphasize reducing whenever you can. The simpler the numbers you work with, the less chance of making a computation error. (Mr. Saxon then goes on to give examples that are not reduced. Shame on him for not showing it both ways!)
We also need to revisit the section on Ratio and Proportion (L 54) to remember that rates are ratios and we can use the principle of cross multiplication for equivalent or proportional ratios. Make sure you are confident about how and when to cross multiply. Only for equivalent fractions!
Finally, we looked at rate again in The Distance Problem (L 58). Note in example 58.1 that both rates are written set up with a 1 in the fraction. You could use an infinite number of proportional rates that are fractions that are equivalent (proportional) to this. If I can drive 40 miles in 1 hour, then I can drive 200 miles in 5 hours or 20 miles in 0.5 hours ( ½ hour or 30 minutes). I can also drive 8 miles in 12 minutes. I can even drive 13.33333 (repeating decimal!) miles in 1/3 hour or 20 minutes. Any number can be the numerator or denominator in a fraction. Usually in textbooks, it works out evenly to a whole number. In real life, that is not so.
Students: It is helpful to do simple versions of these types of problems repeatedly in your head, just to prove that they work. For example, if I can walk 3 miles in 1 hour, I can walk 9 miles in 3 hours. I can walk 1 mile in 20 minutes. I can walk 1.5 miles in ½ hour.
NOW we get to Lesson 63 on rates that change. This is important as an applied skill. After all, real life is rarely constant! I get tired, so I walk more slowly. I practice multiplication facts and I can solve more math problems in a minute than I could at the beginning.
It is important to set the problem up in WRITING and SHOW EVERY STEP.
Especially in a word problem, you have to read carefully. READ ALOUD so that you know what is going on.
Then read your STEPS aloud so you know what you did.
Finally, does your answer MAKE SENSE?
Now let’s look at actual problems. Keep in mind that each one requires careful reading.
In Lesson 63, Practice a. confused some students because the answer is NOT a whole number.
120 miles/ 3 hours = 40 MPH
Increase that speed by 10 MPH and now you are going 50 MPH or 1 hour/ 50 miles
Jim has to drive 120 miles home, so you set up the equation:
120 miles x 1 hour/ 50 miles = hours to home
Cancel the miles and you have 120/50 hours. REDUCE! This is 12/5 hours, but this is a real life situation where we would not want to leave the answer as an improper fraction so we can figure it is
2 and 2/5 hours or 2.4 hours or 2 hours and 24 minutes.
In the real world, we would use the final way of saying our time.
Remember: an hour has 60 minutes, not 100!
This required a lot of steps and careful computation and writing.
(Note: Could you use cross multiplication instead to solve this? Of course! 120/H = 50/1 cross multiplies to 120 x 1 which equals 50 x H. Multiply both sides by the RECIPROCAL of 50 which is 1/50 or DIVIDE both sides by 50 and you are back to the same problem as above.)
Problem 2 inLesson 64 can be SET UP carefully and solved this way:
First Rate A:
HOW FAST? The column went 12 miles in 4 hours which is 3 miles in 1 hour which is 3 MPH
HOW FAR? 12 miles
HOW LONG? 4 hours
Faster Rate B:
HOW FAST? 2 MPH more than the first (3 + 2) = 5 MPH or 1 hour/5 miles
HOW FAR? 52 miles is the total trip minus the 12 they already went= 40 miles
HOW LONG? 40 miles x 1 hour/5 miles which is 40/5 hours which is 8 hours for the second part of the trip
KEEP TRACK of your work. DRAW a picture, but also WRITE DOWN every STEP!
Problem 4 in Lesson 65 can be SET UP carefully and solved:
Sunshine Rate A:
5 kilometers in 1 ½ hours or
5 kilometers in 3/2 hours or
10 kilometers in 3 hours or
10/3 kilometers in 1 hour or
3 1/3 kilometers in 1 hour
Rain Rate B
4 kilometers in 2 hours
2 kilometers in 1 hour
>
> How much slower is the Rain Rate? (That’s what decrease means.)
Changing Rates Lesson 63
Rate (L 34) is first introduced on page 113. We know that rates are fractions. Unfortunately, it is not explicitly stated that you should set a rate up as REDUCED fraction, but you can see in the examples for this lesson that each rate is converted to an equivalent fraction with a 1 in the numerator or a 1 in the denominator. We always write both rates because we could need either one, depending upon the word problem, but we will look at this again in Practice 63a.
You do get a bit of help in Rate Word Problems (L 36) with the “Note” in the middle of the page saying that you can reduce rates, if you want to. For students, it is better to reduce them. I constantly emphasize reducing whenever you can. The simpler the numbers you work with, the less chance of making a computation error. (Mr. Saxon then goes on to give examples that are not reduced. Shame on him for not showing it both ways!)
We also need to revisit the section on Ratio and Proportion (L 54) to remember that rates are ratios and we can use the principle of cross multiplication for equivalent or proportional ratios. Make sure you are confident about how and when to cross multiply. Only for equivalent fractions!
Finally, we looked at rate again in The Distance Problem (L 58). Note in example 58.1 that both rates are written set up with a 1 in the fraction. You could use an infinite number of proportional rates that are fractions that are equivalent (proportional) to this. If I can drive 40 miles in 1 hour, then I can drive 200 miles in 5 hours or 20 miles in 0.5 hours ( ½ hour or 30 minutes). I can also drive 8 miles in 12 minutes. I can even drive 13.33333 (repeating decimal!) miles in 1/3 hour or 20 minutes. Any number can be the numerator or denominator in a fraction. Usually in textbooks, it works out evenly to a whole number. In real life, that is not so.
Students: It is helpful to do simple versions of these types of problems repeatedly in your head, just to prove that they work. For example, if I can walk 3 miles in 1 hour, I can walk 9 miles in 3 hours. I can walk 1 mile in 20 minutes. I can walk 1.5 miles in ½ hour.
NOW we get to Lesson 63 on rates that change. This is important as an applied skill. After all, real life is rarely constant! I get tired, so I walk more slowly. I practice multiplication facts and I can solve more math problems in a minute than I could at the beginning.
It is important to set the problem up in WRITING and SHOW EVERY STEP.
Now let’s look at actual problems. Keep in mind that each one requires careful reading.
In Lesson 63, Practice a. confused some students because the answer is NOT a whole number.
Problem 2 in Lesson 64 can be SET UP carefully and solved this way:
First Rate A:
Faster Rate B:
Problem 4 in Lesson 65 can be SET UP carefully and solved:
Sunshine Rate A:
Rain Rate B
- 4 kilometers in 2 hours
- 2 kilometers in 1 hour
>> How much slower is the Rain Rate? (That’s what decrease means.)
- 3 1/3 kilometers/hour – 2 kilometers/hour = 1 1/3 kilometers/hour decrease
>> The actual math is EASY. Knowing WHAT to do WHEN requires some thinking!